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TORSIONAL VIBRATIONS OF DOUBLY-SYMMETRIC THIN-WALLE D
I-BEAMS RESTING ON WINKLER-PASTERNAK FOUNDATION
USING DYNAMIC MATRIX METHOD
A. SAI KUMAR 1 & K. SRINIVASA RAO 2 1Research Scholar, Department of Mechanical Engineering, Vasavi College of Engineering (Autonomous),
Ibrahimbagh, Hyderabad, Telangana, India 2Associate Professor, Department of Mechanical Engineering, Vasavi College of Engineering (Autonomous),
Ibrahimbagh, Hyderabad, Telangana, India
ABSTRACT
The present work deals with Dynamic Stiffness Analysis of free torsional vibration of doubly
symmetric thin-walled beams of open section and resting on Winkler-Pasternak elastic foundation. A new
dynamic stiffness matrix is developed in this paper which includes the effects of warping and Winkler-
Pasternak foundation on its frequencies of vibration. The resulting transcendental frequency equations for
all classical and various special boundary conditions are solved for thin-walled beams of open cross section
for varying values of warping Winkler and Pasternak foundation parameters on its frequencies of
vibration.
A new MATLAB computer program has been developed based on the dynamic stiffness matrix
approach to solve the highly transcendental frequency equations and to accurately determine the torsional
natural frequencies for all classical and various special boundary conditions. The MATLAB code
developed consists of a master program based on modified BISECTION method and to call specific
subroutines to set up the dynamic stiffness matrix to perform various parametric calculations. Numerical
results for natural frequencies for various values of warping and Winkler and Pasternak foundation
parameters are obtained and presented in graphical form showing their parametric influence clearly.
KEYWORDS: Warping, Dynamic Stiffness Matrix, Winkler-Pasternak Foundation, MATLAB, Bisection
Method
Received: Dec 29, 2015; Accepted: Jan 07, 2016; Published: Jan 19, 2016; Paper Id.: IJCSEIERDFEB20164
INTRODUCTION
The problem of a beam (or plate) on an elastic foundation is important in both the civil and mechanical
engineering fields, since it constitutes a practical idealization for many problems (e.g. the behavior of a shaft
within a hydro-dynamically lubricated bearing, a floating body on the water, etc.). The concept of beams and slabs
on elastic foundations has been extensively used by geotechnical, pavement and railroad engineers for foundation
design and analysis.
Original A
rticle International Journal of Civil, Structural, Environmental and Infrastructure Engineering Research and Development(IJCSEIERD) ISSN(P): 2249-6866; ISSN(E): 2249-7978 Vol. 6, Issue 1, Feb 2016, 31-50 © TJPRC Pvt. Ltd.
32
Impact Factor (JCC): 5.9234
Figure a: Grillage Foundation
The analysis of structures resting on elastic foundations is usually based on a relatively simple model of the
foundation’s response to applied loads. A simple representation of elastic foundation was introduc
[1]. The Winkler model (one parameter model), which has been originally developed for the analysis of railroad tracks, is
very simple but does not accurately represents the characteristics of many practical foundations.
Figure 1: Deflections of Winkler
In order to eliminate the deficiency of Winkler model, improved theories have been introduced on refinement of
Winkler’s model, by visualizing various types of interconnectio
springs [1] (Filonenko-Borodich (1940) [2]; Hetényi (1946) [3]; Pasternak (1954) [4]; Kerr (1964) [5]). These theories
have been attempted to find an applicable and simple model of representation of found
Winkler model shortcomings improved versions [6] [7] have been developed. A shear layer is introduced in the Winkler
foundation and the spring constants above and below this layer is assumed to be different as per this formula
following figure shows the physical representation of the Winkler
The vibrations of continuously
the design of aircraft structures, base frames for rotating machinery, railroad tracks, etc. Quite a good amount of literatur
exists on this topic, and valuable practical methods for the analysis of beams on elastic foundation have been su
11]. Kameswara Rao et al [12-14] studied the problem of torsional vibration of long, thin
resting on Winkler-type elastic foundations using exact, finite element and approximate expressions for torsional frequency
of a thin-walled beams and subjected to a time
A. Sai Kumar & K. Srinivasa Rao
Foundation Figure b: Mat Foundation under Large Storage Tanks
structures resting on elastic foundations is usually based on a relatively simple model of the
foundation’s response to applied loads. A simple representation of elastic foundation was introduc
[1]. The Winkler model (one parameter model), which has been originally developed for the analysis of railroad tracks, is
very simple but does not accurately represents the characteristics of many practical foundations.
: Deflections of Winkler Foundation under Uniform Pressure
In order to eliminate the deficiency of Winkler model, improved theories have been introduced on refinement of
Winkler’s model, by visualizing various types of interconnections such as shear layers and beams along the Winkler
Borodich (1940) [2]; Hetényi (1946) [3]; Pasternak (1954) [4]; Kerr (1964) [5]). These theories
have been attempted to find an applicable and simple model of representation of foundation medium. To overcome the
Winkler model shortcomings improved versions [6] [7] have been developed. A shear layer is introduced in the Winkler
foundation and the spring constants above and below this layer is assumed to be different as per this formula
following figure shows the physical representation of the Winkler-Pasternak model.
Figure 2: Winkler-Pasternak Model
The vibrations of continuously-supported finite and infinite beams on elastic foundation has wide
the design of aircraft structures, base frames for rotating machinery, railroad tracks, etc. Quite a good amount of literatur
exists on this topic, and valuable practical methods for the analysis of beams on elastic foundation have been su
14] studied the problem of torsional vibration of long, thin-walled beams of open sections
type elastic foundations using exact, finite element and approximate expressions for torsional frequency
walled beams and subjected to a time-invariant axial compressive force.
A. Sai Kumar & K. Srinivasa Rao
NAAS Rating: 3.01
Storage Tanks
structures resting on elastic foundations is usually based on a relatively simple model of the
foundation’s response to applied loads. A simple representation of elastic foundation was introduced by Winkler in 1867
[1]. The Winkler model (one parameter model), which has been originally developed for the analysis of railroad tracks, is
very simple but does not accurately represents the characteristics of many practical foundations.
Foundation under Uniform Pressure q
In order to eliminate the deficiency of Winkler model, improved theories have been introduced on refinement of
ns such as shear layers and beams along the Winkler
Borodich (1940) [2]; Hetényi (1946) [3]; Pasternak (1954) [4]; Kerr (1964) [5]). These theories
ation medium. To overcome the
Winkler model shortcomings improved versions [6] [7] have been developed. A shear layer is introduced in the Winkler
foundation and the spring constants above and below this layer is assumed to be different as per this formulation. The
supported finite and infinite beams on elastic foundation has wide applications in
the design of aircraft structures, base frames for rotating machinery, railroad tracks, etc. Quite a good amount of literature
exists on this topic, and valuable practical methods for the analysis of beams on elastic foundation have been suggested. [8-
walled beams of open sections
type elastic foundations using exact, finite element and approximate expressions for torsional frequency
Torsional Vibrations of Doubly-Symmetric Thin-Walled I-Beams Resting on 33 Winkler-Pasternak Foundation Using Dynamic Matrix Method
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It is well known that a dynamic stiffness matrix is mostly formed by frequency-dependent shape functions which
are exact solutions of the governing differential equations. It overcomes the discretization errors and is capable of
predicting an infinite number of natural modes by means of a finite number of degrees of freedom. This method has been
applied successfully for many dynamic problems including natural vibration. A general dynamic-stiffness matrix of a
Timoshenko beam for transverse vibrations was derived including the effects of rotary inertia of the mass, shear distortion,
structural damping, axial force, elastic spring and dashpot foundation [15]. Analytical expressions were derived for the
coupled bending-torsional dynamic stiffness matrix terms of an axially loaded uniform Timoshenko beam element [16-20]
and also a dynamic stiffness matrix is derived based on Bernoulli–Euler beam theory for determining natural frequencies
and mode shapes of the coupled bending-torsion vibration of axially loaded thin-walled beams with mono-symmetrical
cross sections, by using a general solution of the governing differential equations of motion including the effect of warping
stiffness and axial force [21] and [22]. Using the technical computing program Mathematica, a new dynamic stiffness
matrix was derived based on the power series method for the spatially coupled free vibration analysis of thin-walled curved
beam with non-symmetric cross-section on Winkler and also Pasternak types of elastic foundation [23] and [24]. The free
vibration frequencies of a beam were also derived with flexible ends resting on Pasternak soil, in the presence of a
concentrated mass at an arbitrary intermediate abscissa [25]. The static and dynamic behaviors of tapered beams were
studied using the differential quadrature method (DQM) [26] and also a finite element procedure was developed for
analyzing the flexural vibrations of a uniform Timoshenko beam-column on a two-parameter elastic foundation [27].
Though many interesting studies are reported in the literature [8-27], the case of doubly-symmetric thin-walled
open section beams resting on Winkler–Pasternak foundation is not dealt sufficiently in the available literature to the best
of the author’s knowledge.
In view of the above, the present work deals with dynamic stiffness analysis of free torsional vibration of doubly
symmetric thin-walled beams of open section and resting on Winkler-Pasternak elastic foundation. A new dynamic stiffness
matrix (DSM) is developed which includes the effects of warping and Winkler-Pasternak foundation on its frequencies of
vibration. The resulting transcendental frequency equations for all classical and various special boundary conditions are
solved for thin-walled beams of open cross section for varying values of warping and Winkler, Pasternak foundation
parameters on its frequencies of vibration.
A new MATLAB computer program is developed based on the dynamic stiffness matrix approach to solve the
highly transcendental frequency equations for all classical and various special boundary conditions. The MATLAB code
developed consists of a master program based on modified BISECTION method and to call specific subroutines to set up
the dynamic stiffness matrix to perform various parametric calculations. Numerical results for natural frequencies for
various values of warping and Winkler and Pasternak foundation parameters are obtained and presented in graphical form
showing their parametric influence clearly.
NOMENCLATURE
Table 1 �� St. Venant’s torsion T� warping torsion T� Total Non-Uniform Torsion k Modulus Of Subgrade Reaction P Pressure
34 A. Sai Kumar & K. Srinivasa Rao
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Table 1: Contd., �� Shear Modulus ∅ Angle Of Twist G Modulus Of Rigidity Shear Constant M Twisting Moment In Each Flange h Distance Between The Center Lines Of The Flanges �� Moment Of Inertia Of Flange About Its Strong Axis u Lateral Displacement Of The Flange Centerline Warping Constant E Young’s Modulus � Mass Density Of The Material Of The Beam �� Polar Moment Of Inertia � Winkler Foundation Stiffness �� Pasternak Layer Stiffness Z Distance Along The Length Of The Beam � Torsional Natural Frequency K Non-Dimensional Warping Parameter �� Non-Dimensional Pasternak Foundation Parameter �� Non-Dimensional Winkler Foundation Parameter � Non-dimensional frequency parameter �(�) Variation Of Angle Of Twist ∅
FORMULATION AND ANALYSIS
Consider a long doubly-symmetric thin-walled beam of open-section of length L and resting on a Winkler-
Pasternak type elastic foundation of Winkler torsional stiffness (� )and Pasternak layer stiffness(��). The beam is
undergoing free torsional vibrations. The corresponding differential equation of motion can be written as:
� ��∅��� − �� + �� �!∅��! + � ∅ − ��� �!∅�"! = 0 (1)
For free torsional vibrations, the angle of twist ∅(�, &) can be expressed in the form,
∅(�, &) = '(�)()*" (2)
In which '(�) is the modal shape function corresponding to each beam torsional natural frequency �.
The expression for '(�) which satisfies Eq. (1) can be written as
'(�) = , -./ 01 + 2 /34 01 + -./ℎ 61 + 7 /34ℎ 61 (3)
In which 61 849 01 are the positive, real quantities given by
01, 61
= :∓(<!=>?!)=@(<!=>?!)!=A(B!C>D!)E (4)
�E = FGHIJ!KHD L
�� = :<? J!KHD
Torsional Vibrations of Doubly-Symmetric Thin-Walled I-Beams Resting on 35 Winkler-Pasternak Foundation Using Dynamic Matrix Method
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�� = :<D J�KHD
�A = FMN?*!J�KHD L (5)
From Eq. (4) we have the following relation between 61 849 01
(01)E = (61)E + �E − ��E (6)
Knowing 6 849 0, the frequency parameter λ can be evaluated using the relation
�E = (61)(01) + ��E (7)
The four arbitrary constants A, B, C and D in Eq. (3) can be determined from the boundary conditions of the
beam. For any single-span beam, there will be two boundary conditions at each end and these four conditions then
determine the corresponding frequency and mode shape expressions.
DYNAMIC STIFFNESS MATRIX
In order to proceed further, we must first introduce the following nomenclature. The variation of angle of twist ∅
with respect to z is denoted by �(�). The flange bending moment and the total twisting moment are given by O(�) and P(�). Considering clockwise rotations and moments to be positive, we have
�(�) = 9∅ 9�, ℎO(�) = −� 9E∅ 9�R⁄⁄ (8)
P(T) = −�� �UV�WU + �X �V�� (9)
Where E� is termed as the warping rigidity of the section,
� = NYZ!E (10)
Consider a uniform thin-walled I-beam element of length L as shown in the Figure. 3. By combining the Eq. (3)
and Eq. (8), the end displacements, ∅(0)and �(0) and end forces ℎO(0) and P(0), of the beam at � = 0, can be expressed
as
[ ∅(0)�(0)ℎO(0)P(0) \
= ] 1 0 1 00 6 0 0��6E0 0��60E ��0E0 0−��6E0_ `,27a (11)
Eq. (3.28) can be abbreviated as follows
b(0) = c(0)d (12)
In a similar manner, the end displacements, ∅(1)and �(1) and end forces ℎO(1) and P(1), of the beam at � = 1,
can be expressed as
36
Impact Factor (JCC): 5.9234
b(1) = c(1)d
Where
c(1) = [ ∅�1���1�5O�1�P�1� \
edfg # e, 2 7f
hc�1�i # jkk
l - /�6/��6E-���60E/6-��6E/��60E-
In which
- # -./ 01, / # /34 01, # -./
Figure 3
The equation relating the end forces and displacements can be written as
[ P�0�5O�0�P�1�5O�1�\
# jkkkl 0 ��60E��6E���60E/��6E-
0��60E-��6E/ ∗ hdi [∅�0���0�∅�1���1�\
A. Sai Kumar & K. Srinivasa Rao
c0c���0E���6E0c0���0Ec���6E0noo
p
-./5 61, c # /345 61
3: Differential Element of Thin wall I Section Beam
The equation relating the end forces and displacements can be written as
0 ���6E0��0E���6E0c���0E0���6E0���0Ec noo
op
A. Sai Kumar & K. Srinivasa Rao
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(13)
(14)
(15)
(16)
(17)
(18)
Torsional Vibrations of Doubly-Symmetric Thin-Walled I-Beams Resting on 37 Winkler-Pasternak Foundation Using Dynamic Matrix Method
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By eliminating the integration constant vector U and designating the left end element as I and the right end as j,
the final equation relating the end forces and displacements can be written as
qrsrt P) ��⁄5Ou ��⁄Pu ��⁄5Ou ��⁄ vrw
rx # ]yzz yzE yzR yzAyEz yEE yER yEAyRz yRE yRR yRAyAz yAE yAR yAA_ [{)�){u�u
\ (19)
Eq. (19) is symbolically written as
e|f # hyiedf (20)
In the Eq. (20) the matrix hyiis the ‘exact’ element dynamic stiffness matrix (DSM), which is also a square matrix.
The elements of hyiare
yzz # }�6E + 0E��6c- + 0/�
yzE # �}h�6E � 0E��1 � -� + 260c/i yzR # �}�6E + 0E��6/ + 0c�
yzA # �}�6E + 0E�� � -�
yEE # ��} 60⁄ ��6E + 0E��6c- � 0/�
yEA # �} 60⁄ ��6E + 0E��6c � 0/�
yER # �yzA
yRR # yzz
yRA # �yzE
yAA # yEE
and
} # 60 h260�1 � - + �0E � 6E�c/i⁄ (21)
Using the element dynamic stiffness matrix defined by Eq. (20), one can easily set up the general equilibrium
equations for multi-span thin-walled beams, adopting the usual finite element assembly methods. Introducing the boundary
conditions, the final set of equations can be solved for eigenvalues by setting up the determinant of their matrix to zero
METHOD OF SOLUTION
Denoting the modified dynamic stiffness matrix as [J], we state that
9(&|y| # 0 (22)
The above equation yields the frequency equation of continuous thin-walled beams in torsion resting on Winkler-
Pasternak type foundation. It can be noted that above equation is highly transcendental, the roots of equation can, therefore,
be obtained by applying the bisection method using MATLAB code on a high-speed digital computer.
38 A. Sai Kumar & K. Srinivasa Rao
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A new MATLAB code [ANNEX A] was developed based on bisection method, which consists of master program
i.e. (code.m [ANNEX A]) and to call specific subroutines i.e. (FCT.m [ANNEX A]) to perform various parametric
calculations and was published in MATLAB Central official online library [29] which was cited and referred by few
researchers.
These are some of the key highlights of the new MATLAB code
• Primarily, it solves almost any given linear, non-linear & highly-transcendental equations.
• Additional key highlight of this code is, the equation whose roots are to be found, can be defined separately in an
“.m” file, which facilitates to solve multi-variable (example'E + �E + /34 ' + -./ � # 0) highly-transcendental
equations of any size, where the existing MATLAB codes fail to compute.
• This code is made robust in such a manner that it can automatically save and write the detailed informationsuch as
no. of iterations; the corresponding values of the variables and the computing time are automatically saved, into a
(.txt) file format, in a systematic tabular form.
• This code has been tested on MATLAB 7.14 (R2012a) [30] for all possible types of equations and proved to be
accurate.
Exact values of the frequency parameter λ for various boundary conditions of thin-walled open section beam are
obtained and the results are presented both in tabular and graphical form in this paper for varying values of warping,
Winkler foundation and Pasternak foundation parameters.
RESULTS AND DISCUSSIONS
The approach developed in this paper can be applied to the calculation of natural torsional frequencies and mode
shapes of multi-span doubly symmetric thin-walled beams of open section such as beams of I-section. Beams with non-
uniform cross-sections also can be handled very easily as the present approach is almost similar to the finite element
method of analysis but with exact displacement shape functions. All classical and non-classical (elastic restraints) boundary
conditions can be incorporated in the present model without any difficulty.
In the following, six common type and four new types of beams will be identified by a compound objective which
describes the end conditions at (z = 0 and z = L). They are
• Simply-supported beam
The boundary conditions for this problem can be written as
� ∅ = 0O = 0� 8& ' = 0 � ∅ = 0O = 0� 8& ' = 1 (23)
Figure 4: Simply-Supported Beam resting on Winkler-Pasternak Foundation
Torsional Vibrations of Doubly-Symmetric Thin-Walled I-Beams Resting on 39 Winkler-Pasternak Foundation Using Dynamic Matrix Method
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Considering a one element solution and applying these boundary conditions to Eq. (19) gives
| yEEyAA − yEAyAE| = 0 (24)
This gives,
(} 60)(6E + 0E)E⁄ /34(01) ∗ /34ℎ(61) = 0 (25)
As H and (6E + 0E)E are in general, non-zero. The frequency equation for the simply supported beam can,
therefore, be written as,
/34(01) ∗ /34ℎ(61) = 0 (26)
• Fixed-end beam
The boundary conditions for this problem can be written as
�∅ = 0� = 0� 8& ' = 0 �∅ = 0� = 0� 8& ' = 1 (27)
Figure 5: Fixed-end Beam Resting on Winkler-Pasternak Foundation
Considering a one element solution and applying these boundary conditions to Eq. (19) gives
(1 − -./ℎ(61) -./(01)) + (�!C�!�)E�� ∗ /34ℎ(61) /34(β1) = 0 (28)
• Beam free at both the ends
The boundary conditions for this problem can be written as
� P = 0O = 0� 8& ' = 0 � P = 0O = 0� 8& ' = 1 (29)
Figure 6: Beam Free at Both ends and resting on Winkler-Pasternak Foundation
Considering a one element solution and applying these boundary conditions to Eq. (19) gives
|y| = 0 (30)
This gives,
40 A. Sai Kumar & K. Srinivasa Rao
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�1 � -./5�61� -./�01�� � ���C���E��U�U� ∗ /34ℎ(61) /34(01) = 0 (31)
• Beam fixed at one end, simply-supported at other end
The boundary conditions for this problem can be written as
�∅ = 0� = 0� 8& ' = 0 � ∅ = 0O = 0� 8& ' = 1 (32)
Figure 7: Beam Fixed at one end, Simply-Supported at Other End and Resting on Winkler-Pasternak Foundation
Considering a one element solution and applying these boundary conditions to Eq. (19) gives
| yEE| = 0 (33)
This gives,
(} 60)(6E + 0E)⁄ (6c- − 0/) = 0 (34)
As H and (6E + 0E) are in general non-zero. The frequency equation for the simply supported beam can,
therefore, be written as,
6 &84(01) − 0 &84ℎ(61) = 0 (35)
• Beam fixed at one end, free at other end
The boundary conditions for this problem can be written as
�∅ = 0� = 0� 8& ' = 0 � P = 0O = 0� 8& ' = 1 (36)
Figure 8: Beam Fixed at One End, Free at Other End and Resting on Winkler-Pasternak Foundation
Considering a one element solution and applying these boundary conditions to Eq. (19) gives
| yRRyAA − yRAyAR| = (37)
This gives,
(��=��)�!�! -./ℎ(61) -./(01) + (�!C�!)�� /34ℎ(61) /34(01) + 2 = 0 (38)
Torsional Vibrations of Doubly-Symmetric Thin-Walled I-Beams Resting on 41 Winkler-Pasternak Foundation Using Dynamic Matrix Method
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• Beam simply-supported at one end, free at other end
The boundary conditions for this problem can be written as
� ∅ = 0O = 0� 8& ' = 0 � P = 0O = 0� 8& ' = L (39)
Figure 9: Beam Simply-Supported at one end, Free at Other End and Resting on Winkler-Pasternak Foundation
Considering a one element solution and applying these boundary conditions to Eq. (19) gives
|yEE(yRRyAA − yRAyAR) − yER(yREyAA − yRAyER) + yEA(yREyAR − yRRyAE)| = 0 (40)
This gives,
0R &84(01) − 6R &84ℎ(61) = 0 (41)
• Guided-end beam
The boundary conditions for this problem can be written as
�� = 0P = 0� 8& ' = 0 �� = 0P = 0� 8& ' = 1 (42)
Figure 10: Beam Guided at both the ends and resting on Winkler-Pasternak Foundation
Considering a one element solution and applying these boundary conditions to Eq. (19) gives
| yRRyzz − yzRyRz| = 0 (43)
This gives,
(})(60)(/34(01) ∗ /34ℎ(61) = 0 (44)
As H and (6E + 0E)E are in general, non-zero. The frequency equation for the simply supported beam can,
therefore, be written as,
(/34(01) ∗ /34ℎ(61) = 0 (45)
• Beam guided at one end, free at other end
The boundary conditions for this problem can be written as
42 A. Sai Kumar & K. Srinivasa Rao
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�� # 0P = 0� 8& ' = 0 � P = 0O = 0� 8& ' = 1 (46)
Figure 11: Beam Guided at One End, Free at Other End and Resting on Winkler-Pasternak Foundation
Considering a one element solution and applying these boundary conditions to Eq. (19) gives
|yzz(yRRyAA − yRAyAR) − yzR(yRzyAA − yRAyAz) + yzA(yRzyAR − yRRyAz)| = 0 (47)
This gives,
6R &84(01) + 0R &84ℎ(61) = 0 (48)
• Beam guided at one end, simply supported at one end
The boundary conditions for this problem can be written as
�� = 0P = 0� 8& ' = 0 � ∅ = 0O = 0� 8& ' = 1 (49)
Figure 12: Beam Guided at One End, Simply Supported at other End and Resting on Winkler-Pasternak Foundation
Considering a one element solution and applying these boundary conditions to Eq. (19) gives
| yAAyzz − yzAyAz| = 0 (50)
This gives,
E�!�!(��=��) -./ℎ(61) -./(01) + 1 = 0 (51)
• Beam guided at one end, clamped at other end
The boundary conditions for this problem can be written as
�� = 0P = 0� 8& ' = 0 �∅ = 0� = 0� 8& ' = 1 (52)
Figure 13: Beam Guided at One and, fixed at Other End and Resting on Winkler-Pasternak Foundation
Torsional Vibrations of Doubly-Symmetric Thin-Walled I-Beams Resting on 43 Winkler-Pasternak Foundation Using Dynamic Matrix Method
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Considering a one element solution and applying these boundary conditions to Eq. (19) gives
|yzz| # 0 (53)
This gives,
0 &84(01) + 6 &84ℎ(61) = 0 (54)
The general dynamic stiffness matrix defined by Eq. (33) and Eq. (47) of Ref. [15], for Euler-Bernoulli beam is
observed to be same as Eq. (19) and Eq. (21), but for only difference that the axial force was included in the Eq. (33) and
Eq. (47) of Ref. [15], in defining the roots 6 849 0 where as in the present paper the axial force was not included in the
Eq. (19) and Eq. (21).
The first order approximation equations (Eq. 11, Eq. 18, Eq. 22, Eq. 25, Eq. 27, and Eq. 30) of Ref. [31], for
torsional vibrations of uniform doubly symmetric thin walled open cross section are observed to be same as equations (Eq.
(25), Eq. (28), Eq. (31), Eq. (35), Eq. (38) and Eq. 41)).
The transcendental frequency equations (Eq. (4a), Eq. (4b), Eq. (4e) and Eq. (4f)) of Ref. [32], for generally
restrained beams are observed to be same as equations (Eq. (45), Eq. (48), Eq. (51) and Eq. (54)) of the present paper, but
for only difference that the roots 6 849 0 are considered to be equal and have same sign and defined as��.
The equations for the fixed-end beam and simply supported beam are solved for values of warping parameter 0 ≤ � ≤ 20 and for various values of Winkler foundation parameter 0 ≤ � ≤ 10 000 000 and values of Pasternak
foundation parameter 0 ≤ �� ≤ 2.5.
Table 1: Numerical Comparisons Frequency Parameters for a Simply Supported Beam Fully Supported on a Winkler
��, �� 0 Approx.
0 exact
0.5 Approx.
0.5 exact
1 Approx.
1 exact
2.5 Approx.
2.5 exact
0 3.14159 3.1415930 3.4767 3.180679 3.7360 3.2183142 4.2970 3.3240213 1 3.1496 3.1496248 3.48267 3.1883879 3.74078 3.2257872 4.30016 3.3308077
100 3.74836 3.7483635 3.9608 3.7715736 4.10437 3.7943618 4.58239 3.8603697 10000 10.0244 10.0242642 10.036 10.0255880 10.048 10.0267120 10.084 10.0303817
1000000 31.6235 31.6235460 31.6239 31.6235850 31.624 31.6236240 31.625 31.6237412 -Pasternak foundation between finite element method [Ref. 25, 26, 27] and available results
The above Table depicts a comparison between the finite element and exact results for the frequency parameters
of asimply-supported beam fully supported on a Winkler-Pasternak foundation. The results obtained by the dynamic
stiffness matrix approach agree very closely, with the solutions computed from the frequency equations reported in
References [25] [26] and [27].
Furthermore, the equations for all the BC’s are also solved for values of warping parameter 0 ≤ � ≤ 20 and for
various values of Winkler foundation parameter 0 ≤ � ≤ 15 and values of Pasternak foundation parameter 0 ≤ �� ≤ 1
and are presented in graphical form for the first three modes.
The following Figure 14 shows the variation of frequency parameter with foundation parameters� 849 γ�, for
simply supported beam, one end fixed and other end free beam.
44
Impact Factor (JCC): 5.9234
Figure 14: Plot for Parameter for values of Pasternak
The influences of the foundation
conditions is shown in the Figure 14, The figures indicate that the stability parameter increases as the overall stiffness of
the beam-foundation system increases. The overall
support stiffness, the foundation stiffness and the flexural rigidity of the beam. It is known that the flexural rigidity of
beam increases as � increases. It is obvious that th
foundation system increases.
Figure 15 shows the variation of frequency parameter with warping parameter K, for simply supported beam,
fixed-beam, one end fixed and other end free beam, o
A close look at the results presented in Figure 15 clearly reveals that the effect of an increase in warping
parameter K is to drastically decrease the fundamental frequency
foundation is found to increase the frequency of vibration especially for the first few modes. However, this influence is
seen to be quite negligible on the modes higher than the third.
Figure 15: Plot for Influence of Pasternak Foundation Parameter
Furthermore, the equations for guided end condition are also solved for values of warping parameter
and for various values of Winkler foundation parameter 0 � �� � 1 are presented in graphical form for the first three modes.
The influences of the foundation parameters
shown in the Figure 16, The figure indicate that the stability parameter increases as the overall stiffness of the beam
foundation system increases. The overall stiffness of the beam
A. Sai Kumar & K. Srinivasa Rao
for Influence of Winkler Foundation Parameter on Frequency for values of Pasternak Foundation Parameter for Various B
The influences of the foundation parameters� 849 ��, on the stability parameter for different supporting
conditions is shown in the Figure 14, The figures indicate that the stability parameter increases as the overall stiffness of
foundation system increases. The overall stiffness of the beam-foundation system is an integrated resultant of the
support stiffness, the foundation stiffness and the flexural rigidity of the beam. It is known that the flexural rigidity of
increases. It is obvious that the frequency parameter increases as the overall stiffness of the beam
Figure 15 shows the variation of frequency parameter with warping parameter K, for simply supported beam,
beam, one end fixed and other end free beam, one end fixed and other end simply supported beam
A close look at the results presented in Figure 15 clearly reveals that the effect of an increase in warping
parameter K is to drastically decrease the fundamental frequency�. Furthermore, can be expected,
foundation is found to increase the frequency of vibration especially for the first few modes. However, this influence is
seen to be quite negligible on the modes higher than the third.
Influence of Warping Parameter on Frequency Parameter Foundation Parameter and Winkler Foundation Parameter for
Furthermore, the equations for guided end condition are also solved for values of warping parameter
and for various values of Winkler foundation parameter 0 � � � 15 and values of Pasternak foundation parameter
are presented in graphical form for the first three modes.
The influences of the foundation parameters� 849 ��, on the stability parameter for guided
shown in the Figure 16, The figure indicate that the stability parameter increases as the overall stiffness of the beam
foundation system increases. The overall stiffness of the beam-foundation system is an integr
A. Sai Kumar & K. Srinivasa Rao
NAAS Rating: 3.01
Frequency Various BC’s
, on the stability parameter for different supporting
conditions is shown in the Figure 14, The figures indicate that the stability parameter increases as the overall stiffness of
foundation system is an integrated resultant of the
support stiffness, the foundation stiffness and the flexural rigidity of the beam. It is known that the flexural rigidity of the
e frequency parameter increases as the overall stiffness of the beam
Figure 15 shows the variation of frequency parameter with warping parameter K, for simply supported beam,
ne end fixed and other end simply supported beam.
A close look at the results presented in Figure 15 clearly reveals that the effect of an increase in warping
. Furthermore, can be expected, the effect of elastic
foundation is found to increase the frequency of vibration especially for the first few modes. However, this influence is
Frequency Parameter for Values for Various BC’s
Furthermore, the equations for guided end condition are also solved for values of warping parameter 0 � � � 20
and values of Pasternak foundation parameter
ity parameter for guided-end conditions is
shown in the Figure 16, The figure indicate that the stability parameter increases as the overall stiffness of the beam-
foundation system is an integrated resultant of the support
Torsional Vibrations of Doubly-Symmetric Winkler-Pasternak Foundation Using Dynamic Matrix Method
www.tjprc.org
stiffness, the foundation stiffness and the flexural rigidity of the guided
parameter increases as the overall stiffness of the beam foundation system increases.
The variation of frequency parameter with foundation parameters
free beam, one end guided and other end simply supported beam are shown graphically.
The plots clearly show that while the Winkler foundation independently increase
vibration for constant values of warping and the Pasternak foundation parameters. Interestingly we can clearly observe
the effect of Pasternak foundation Parameter is to decrease the natural torsional frequency significa
vibration and for constant values of warping and Winkler foundation parameter.
Figure 16: Plot for Influence for values of Pasternak
A close look at the results presented in Figure 17 clearly reveals that the effect of an increase in warping
parameter K is to drastically decrease the fundamental frequency
foundation is found to increase the frequency of vibration especially for the first few modes. However, this influence is
seen to be quite negligible on the modes higher than the third.
Figure 17: Plot For influof Pasternak Foundation Parameter
It can be finally concluded that for an appropriately designing the thin
on continuous elastic foundation, it is very much necessary to model the foundation appropriately considering the Winkler
and Pasternak foundation stiffness values as their combined influence on the natural torsional frequency is quite significa
and hence cannot be ignored.
Symmetric Thin-Walled I-Beams Resting on Dynamic Matrix Method
stiffness, the foundation stiffness and the flexural rigidity of the guided-end beam. It is obvious that the frequency
parameter increases as the overall stiffness of the beam foundation system increases.
uency parameter with foundation parameters� 849 ��, for guided
free beam, one end guided and other end simply supported beam are shown graphically.
The plots clearly show that while the Winkler foundation independently increases the frequency for any mode of
vibration for constant values of warping and the Pasternak foundation parameters. Interestingly we can clearly observe
the effect of Pasternak foundation Parameter is to decrease the natural torsional frequency significa
vibration and for constant values of warping and Winkler foundation parameter.
Influence of Winkler Foundation Parameter on Frequency Paramefor values of Pasternak Foundation Parameter for Guided-End BC’s
A close look at the results presented in Figure 17 clearly reveals that the effect of an increase in warping
parameter K is to drastically decrease the fundamental frequency�. Furthermore, it can be expected, the
foundation is found to increase the frequency of vibration especially for the first few modes. However, this influence is
seen to be quite negligible on the modes higher than the third.
influ ence of Warping Parameter on Frequency Parameter Parameter and Winkler Foundation Parameter for
It can be finally concluded that for an appropriately designing the thin-walled beams of open cross sectio
on continuous elastic foundation, it is very much necessary to model the foundation appropriately considering the Winkler
and Pasternak foundation stiffness values as their combined influence on the natural torsional frequency is quite significa
45
end beam. It is obvious that the frequency
, for guided-clamped beam, guided
s the frequency for any mode of
vibration for constant values of warping and the Pasternak foundation parameters. Interestingly we can clearly observethat
the effect of Pasternak foundation Parameter is to decrease the natural torsional frequency significantly for any mode of
Frequency Parameter BC’s
A close look at the results presented in Figure 17 clearly reveals that the effect of an increase in warping
. Furthermore, it can be expected, the effect of elastic
foundation is found to increase the frequency of vibration especially for the first few modes. However, this influence is
Frequency Parameter for Values for Guided-end BC’s
walled beams of open cross sections resting
on continuous elastic foundation, it is very much necessary to model the foundation appropriately considering the Winkler
and Pasternak foundation stiffness values as their combined influence on the natural torsional frequency is quite significant
46 A. Sai Kumar & K. Srinivasa Rao
Impact Factor (JCC): 5.9234 NAAS Rating: 3.01
CONCLUSIONS
In this paper, a dynamic stiffness matrix (DSM) approach has been developed for computing the natural torsion
frequencies of long, doubly-symmetric thin-walled beams of open section resting on continuous Winkler-Pasternak type
elastic foundation. The approach presented in this thesis is quite general and can be applied for treating beams with non-
uniform cross-sections and also non-classical boundary conditions. A new MATLAB computer program has been
developed based on the dynamic stiffness matrix approach to solve the highly transcendental frequency equations and to
accurately determine the torsional natural frequencies for all classical and various special boundary conditions. Numerical
results for natural frequencies for various values of warping and Winkler and Pasternak-foundation parameters are obtained
and presented in both tabular as well as graphical form showing their parametric influence clearly. From the results
obtained following conclusions are drawn.
• An attempt was made to validate the present formulation of the problem for various boundary conditions. There is
very good agreement between the results, the general dynamic stiffness matrix defined by Eq. (33) and Eq. (47) of
Yung-Hsiang Chen [15], for Euler-Bernoulli beam is observed to be same as Eq. (19) and Eq. (20) in this paper,
but for only difference that the axial force was not included in the present paper.
• Further validation of the model was done by comparing the results obtained for simply supported beams and are
solved for values of various values of Winkler foundation parameter 0 ≤ γ� ≤ 10 000 000 and Pasternak
foundation parameter 0 ≤ γ� ≤ 2.5 and are presented in Table 1. The results compare very well with those from
De Rosa, M. A. and M. J. Maurizio [25] and Yokoyama [26]
• The influences of the foundation parametersγ�, γ� and warping parameter K, on the stability parameter λ for
various supporting conditions are shown in the Figures (14-17). The second foundation parameter γ�, tends to
increase the fundamental frequency for the same Winkler constantγ�. The effect of γ�, may be interpreted in the
following way: A simply supported beam, which is the weakest as far as stability (λ = 3.66) is concerned,
acquires the stability of a beam which is clamped at both ends (λ = 5.24), by increasing the shear parameter of
the foundation especially for the first mode. However, this influence is seen to be quite negligible on the modes
higher than the first. Also, it is found that the effect of an increase in warping parameter K is to drastically
decrease the stability parameterλ.
It can be finally concluded that for an appropriately designing the thin-walled beams of open cross sections resting
on continuous elastic foundation, it is very much necessary to model the foundation appropriately considering the Winkler
and Pasternak foundation stiffness values as their combined influence on the natural torsional frequency is quite significant
and hence cannot be ignored.
Future Work
The Dynamic Stiffness Matrix (DSM) approach could be implemented to multi span beams of open section
resting on various possible types of elastic foundations, including the effects of longitudinal inertia, axial compressive load,
time varying loads and shear deformation. This Dynamic Stiffness Matrix (DSM) approach could also be implemented not
only to beams but also to pipes conveying fluid and carbon Nano-tubes conveying fluid resting on visco-elastic foundation
and three parameter foundation models.
Torsional Vibrations of Doubly-Symmetric Thin-Walled I-Beams Resting on 47 Winkler-Pasternak Foundation Using Dynamic Matrix Method
www.tjprc.org [email protected]
REFERENCES
1. Winkler, E. "Theory of elasticity and strength." Dominicus Prague, Czechoslovakia (1867).
2. Filonenko-Borodich M. M., "Some approximate theories of the elastic foundation." Uchenyie Zapiski Moskovskogo
Gosudarstvennogo Universiteta Mekhanica 46 (1940): 3-18.
3. Hetényi, Miklós, and Miklbos Imre Hetbenyi., “Beams on elastic foundation: theory with applications in the fields of civil and
mechanical engineering”. Vol. 16. University of Michigan Press, 1946.
4. Pasternak, P. L., "On a new method of analysis of an elastic foundation by means of two foundation constants."
Gosudarstvennoe Izdatelstvo Literaturi po Stroitelstvu i Arkhitekture, Moscow (1954).
5. Kerr, Arnold D., "Elastic and viscoelastic foundation models." Journal of Applied Mechanics 31.3 (1964): 491-498.
6. S. C. Dutta en, R. Rana, “A critical review on idealization and modeling for interaction among soil-foundation-structure
system”. Elsevier Science Ltd., pp. 1579-1594, April 2002.
7. Y. H. Wang, L. G. Tham en Y. K. Cheung, “Beams and Plates on Elastic Foundations: a review,” Wiley Inter Science, pp. 174-
182, May 2005.
8. Timoshenko, Stephen P., "Theory of bending, torsion and buckling of thin-walled members of open cross section." Journal of
the Franklin Institute 239.4 (1945): 249-268.
9. Gere, JMt., "Torsional vibrations of beams of thin-walled open section." Journal of Applied Mechanics-Transactions of the
ASME 21.4 (1954): 381-387.
10. Christiano, Paul, and Larry Salmela., "Frequencies of beams with elastic warping restraint." Journal of the Structural
Division 97.6 (1971): 1835-1840.
11. E. J. Sapountzakis, Bars under Torsional loading: a generalized beam approach, ISRN Civil Engineering (2013) 1-39.
12. Rao, C. Kameswara, and S. Mirza, "Torsional vibrations and buckling of thin-walled beams on elastic foundation." Thin-
walled structures 7.1 (1989): 73-82.
13. C. Kameswara Rao and Appala Satyam, "Torsional Vibrations and Stability of Thin-walled Beams on Continuous Elastic
Foundation", AIAA Journal, Vol. 13, 1975, pp. 232- 234.
14. C. Kameswara Rao and S. Mirza., “Torsional vibrations and buckling of thin walled beams on Elastic foundation”, Thin-
Walled Structures (1989) 73-82.
15. Yung-Hsiang Chen., “General dynamic-stiffness matrix of a Timoshenko beam for transverse vibrations”, Earthquake
Engineering and Structural dynamics (1987) 391-402
16. J.R. Banerjee., and F.W. Williams., “Coupled bending-torsional dynamic stiffness matrix of an Axially loaded Timoshenko
beam element”, Journal of Solids Structures, Elsevier science publishers 31 (1994) 749-762
17. P.O. Friberg, “Coupled vibrations of beams-an exact dynamic element stiffness matrix”, International Journal for Numerical
Methods in Engineering 19 (1983) 479-493
18. J.R. Banerjee, “Coupled bending-torsional dynamic stiffness matrix for beam elements”, International Journal for Numerical
Methods in Engineering 28 (1989)1283-1298.
19. Zongfen Zhang and Suhuan Chen, “A new method for the vibration of thin-walled beams”, Computers & Structures 39(6)
(1991) 597-601.
48 A. Sai Kumar & K. Srinivasa Rao
Impact Factor (JCC): 5.9234 NAAS Rating: 3.01
20. J.R. Banerjee and F.W. Williams, “Coupled bending-torsional dynamic stiffness matrix for Timoshenko beam elements”,
Computers & Structures 42(3) (1992)301-310.
21. J.R. Banerjee, “Exact dynamic stiffness matrix of a bending-torsion coupled beam including warping”, Computers &
Structures 59(4) (1996) 613-621.
22. Jun, Li, et al. "Coupled bending and torsional vibration of axially loaded Bernoulli–Euler beams including warping effects."
Applied Acoustics 65.2 (2004): 153-170.
23. Kim, Nam-Il, Chung C. Fu, and Moon-Young Kim. "Dynamic stiffness matrix of non-symmetric thin-walled curved beam on
Winkler and Pasternak type foundations." Advances in Engineering Software 38.3 (2007): 158-171.
24. Nam-Il Kim, Ji-Hun Lee, Moon-Young Kim, “Exact dynamic stiffness matrix of non-symmetric thin-walled beamson elastic
foundation using power series method”, Advances in Engineering Software 36 (2005) 518–532.
25. De Rosa, M. A., and M. J. Maurizi,. "The Influence of Concentrated Masses and Pasternak Soil on the Free Vibrations of Euler
Beams-Exact Solution." Journal of Sound and Vibration 212.4 (1998): 573-581.
26. Yokoyama, T., "Vibrations of Timoshenko beam‐columns on two‐parameter elastic foundations." Earthquake Engineering &
Structural Dynamics 20.4 (1991): 355-370.
27. Hassan, Mohamed Taha, and Mohamed Nassar., "Static and Dynamic Behavior of Tapered Beams on Two-Parameter
Foundation." Vol. 14.(2013): 176-182.
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Business Media, 2008.
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30. MATLAB 7.14 (R2012a)
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Normal Modes of thin-walled open section beams”, journal of aeronautical society of India,1974., pp. 32-41.
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vibration 147.1 (1991): 167-171.
APPENDICES
ANNEX A
MATLAB CODE FOR SOLVING TORSIONAL FREQUENCY EQUATIO NS
Main program function code(a,b) % This code finds a solution to f(x) = 0 % % % % it finds a root given in the continuous function on the interval [a,b], % % where f(a) and f(b) have opposite signs. % % % % INPUT: % % a,b: define the interval over which the met hod is exercised % % tol: is the solution tolerance % % n: is the maximum number of iterations of the algorithm. % % FCT(TOL): deceleration of the function whose sol ution has to be found. % % % % OUTPUT: % % value: is the approximate solution %
Torsional Vibrations of Doubly-Symmetric Thin-Walled I-Beams Resting on 49 Winkler-Pasternak Foundation Using Dynamic Matrix Method
www.tjprc.org [email protected]
% % % USAGE: % % [value] = code(...) to display a solution of th e function % % % %-----------------------------BY: Kumar Sai-------- ------------------------ if nargin < 2 error( 'Incorrect input!!! provide at least two input argu ments' ) end tol=10^-6; n=100; if a > b X = a; a = b; b = X; end if a==b error( 'Input (a) cannot equal (b)' ) end TOL=a; fa = FCT(TOL); TOL=b; fb = FCT(TOL); if fa*fb > 0 error( 'f(a) and f(b) have the same sign' ) end if tol < 0 error( 'tolerance must be a positive number' ) end fileID = fopen( 'results.txt' , 'w' ); fprintf(fileID, 'Solution of Non-Linear Transcendental Freq. Eq.\r\ n' ); fprintf(fileID, '-----------------By: Kumar Sai-----------------\r\ n' ); fprintf(fileID, '---------Number of Iterations obtained---------\r\ n' ); fprintf(fileID, ' I a b c p \r\n ' ); fprintf(fileID, '----------------------------------------------\r\n ' ); tic; I = 1; while I <= n c = (b - a) / 2.0; p = a + c; TOL=p; fp = FCT(TOL); A = [I;a;b;c;p]; fprintf(fileID, '%3u: % 5.7f % 5.7f % 5.7f % 5.7f\r\n' ,A); if I == 1 Result{1} = '-------------------------------------------------- ' ; Result{2} = '--Solution of Non-Linear Transcendental Freq. Eq.- ' ; Result{3} = '-------------------------------------------------- ' ; Result{4} = '-----The value of the Frequency parameter is------ ' ; Result{5} = '-------------------------------------------------- ' ; Result{6} = '----------time elapsed in milliseconds------------ ' ; end if abs(fp) < 1.0e-20 || c < tol Result = char(Result); disp(Result) value = p; t=toc; fprintf(fileID, '--------------------------------------------\r\n' ); fprintf(fileID, '---The value of the Frequency parameter is--\r\n' ); fprintf(fileID, '..............>>> %5.7f <<<...........\r\n' ,value); fprintf(fileID, 'time elapsed in milliseconds: %-10.10f\r\n' ,t*10^3);
50 A. Sai Kumar & K. Srinivasa Rao
Impact Factor (JCC): 5.9234 NAAS Rating: 3.01
fprintf( '------------------ %-10.10f-------------------\n' ,t*10^3); fprintf( '------The value of the Frequency parameter is----- \r\n' ); fprintf( '.................>>> %5.7f <<<...............\r\n' ,value); fclose(fileID); msgbox( 'procedure completed successfully !!' ) break else I = I+1; if fa*fp > 0 a = p; fa = fp; else b = p; end end end end Sub program for simply supported beam
function [F] =FCT(TOL) k2=0; d2=0; G1=0; G2=0; C=(k2+d2+G2); B4=TOL^4; ALP=(sqrt(-C+sqrt((C^2)+4*(B4-G1)))/sqrt(2)); BTA=(sqrt(C+sqrt((C^2)+4*(B4-G1)))/sqrt(2)); F=sinh(BTA)*sin(ALP); end